a) Cho \(S=\frac{1}{31}+\frac{1}{32}+\frac{1}{33}+\frac{1}{60}\)

Chứng minh \(\frac{3}{5}< S< \frac{4}{5}\)

b) Chứng minh \(\frac{1}{41}+\frac{1}{42}+\frac{1}{43}+......+\frac{1}{100}>\frac{7}{10}\)

 Chứng minh rằng :

a) S=\(\frac{3}{1.4}\)+\(\frac{3}{4.7}\)+...+\(\frac{3}{n(n+3)}\)< 1 .

b) Cho : S=\(\frac{1}{2^2}\)+\(\frac{1}{3^2}\)+\(\frac{1}{4^2}\)+...+\(\frac{1}{9^2}\). Chứng minh :\(\frac{2}{5}\)< S <\(\frac{8}{9}\).

   Gợi ý : Chứng minh \(\frac{1}{b}\)-\(\frac{1}{b+1}\)<\(\frac{1}{b^2}\)<\(\frac{1}{b-1}\)-\(\frac{1}{b}\) ( b\(\varepsilon\)\(ℕ\), b > 1 ) .

Cho M = \(\frac{1}{2}\)× \(\frac{3}{4}\)×\(\frac{5}{6}\)× ...×\(\frac{99}{100}\)

Cho N = \(\frac{2}{3}\)× \(\frac{4}{5}\)×\(\frac{6}{7}\)× ...×\(\frac{100}{101}\)

Chứng minh rằng M < N

Tính M . N

Chứng minh rằng M < \(\frac{1}{10}\)

Bài 1: Chứng tỏ các tổng sau không là số tự nhiên:

a. A= \(\frac{1}{2}\)+\(\frac{1}{3}\)+\(\frac{1}{4}\)

b. B= \(\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+...+\frac{1}{8}\)

c. C= \(\frac{3}{10}+\frac{3}{11}+\frac{3}{12}+\frac{3}{13}+\frac{3}{14}\)

Bài 2: Chứng tỏ rằng:

a. A= \(\frac{1}{11}+\frac{1}{12}+\frac{1}{13}+\frac{1}{14}+...+\frac{1}{20}>\frac{1}{2}\)

b. B=\(\frac{1}{50}+\frac{1}{51}+\frac{1}{52}+...+\frac{1}{99}>\frac{1}{2}\)

c. C= \(\frac{1}{10}+\frac{1}{11}+\frac{1}{12}+\frac{1}{100}>1\)

d. D=\(\frac{1}{41}+\frac{1}{42}+\frac{1}{43}+...+\frac{1}{80}>\frac{7}{12}\)

Bài 3: Cho S= \(\frac{1}{31}+\frac{1}{32}+\frac{1}{33}+...+\frac{1}{60}.\)Chứng minh rằng \(\frac{3}{5}< S< \frac{4}{5}\)

Bài 4: Cho B= \(\frac{10n}{5n-3}\), tìm số nguyên n để:

a. B có giá trị nguyên b. B có GTLN

chứng minh :

a,A=1+\(\frac{1}{2^2}\)+\(\frac{1}{3^2}\)+\(\frac{1}{4^2}\)+...+\(\frac{1}{100^2}\)< 2

b,B=1+\(\frac{1}{2}\)+\(\frac{1}{3}\)+\(\frac{1}{4}\)+...+\(\frac{1}{63}\)< 6

c,C=\(\frac{1}{2}\).\(\frac{3}{4}\).\(\frac{5}{6}\).\(\frac{7}{8}\)...\(\frac{9999}{10000}\)< \(\frac{1}{100}\)

Chứng minh rằng :

a)\(\frac{1}{2}\)-\(\frac{1}{4}\)+\(\frac{1}{8}\)-\(\frac{1}{16}\)+\(\frac{1}{32}\)-\(\frac{1}{64}\)<\(\frac{1}{3}\)

b)\(\frac{1}{3}\)-\(\frac{2}{3^2}\)+\(\frac{3}{3^3}\)-\(\frac{4}{3^4}\)+......+\(\frac{99}{3^{99}}\)-\(\frac{100}{3^{100}}\)<\(\frac{3}{16}\)

chứng minh rằng

B=\(\frac{1}{2}\)*\(\frac{3}{4}\)*\(\frac{5}{6}\)*...*\(\frac{199}{200}\).CHỨNG MINH B²<\(\frac{1}{201}\)

C= 1+\(\frac{1}{2}\)+\(\frac{1}{3}\)+\(\frac{1}{4}\)+...+\(\frac{1}{2^{100}-1}\)CHỨNG MINH C<100

CÍU MÌNH CHIỀU NỌP RỒI Ạ

THANK YOU

`1.

a/Chứng minh rằng: \(\frac{1}{41}\)+\(\frac{1}{42}\)+\(\frac{1}{43}\)+...+\(\frac{1}{80}\)>\(\frac{7}{12}\)

b/Chứng minh rằng:: \(\frac{1}{41}\)+\(\frac{1}{42}\)+\(\frac{1}{43}\)+...+\(\frac{1}{60}\)>\(\frac{1}{3}\)

c//Chứng minh rằng:: \(\frac{1}{61}\)+\(\frac{1}{62}\)+\(\frac{1}{62}\)+...+\(\frac{1}{80}\)>\(\frac{1}{4}\)

2.Cho phân số\(\frac{a}{b}\)=\(\frac{1}{6}\)+\(\frac{1}{7}\)+\(\frac{1}{8}\)+\(\frac{1}{9}\)+\(\frac{1}{10}\)+\(\frac{1}{11}\)với a,b thuộc N. Chứng minh rằng a chia hết cho 17

\(\frac{1}{6}\)